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The generalized uncertainty principle provides a promising phenomenological approach to reconciling the fundamentals of general relativity with quantum mechanics. As a result, noncommutativity, measurement uncertainty, and the fundamental theory of quantum mechanics are subject to finite gravitational fields. The generalized uncertainty principle (RGUP) on curved spacetime allows for the imposition of quantum-induced elements on general relativity (GR) in four dimensions. In the relativistic regimes, the determination of a test particle’s spacetime coordinates, 〈xμ〉, becomes uncertain. There exists a specific range of coordinates where the accessibility to 〈(xμ)2〉 is notably limited. Consequently, the spacetime coordinates lack both smoothness and continuity. The quantum-mechanical calculations of the spacetime coordinates are directly linked to their measurement through the expectation value 〈xμ〉. This expectation value is dependent on 〈(xμ)2〉, which is itself limited by a minimum measurable length Δxmin2. A crucial finding presented in this script is the existence of a lower bound for the Hamiltonian, which implies the stability of the quantum nature of spacetime. The direct correlation between Δxmin and −〈g〉μν is a significant discovery, suggesting a proportional relationship. The conjecture is made that the primary metric g carries all essential information regarding spacetime curvature and serves a role akin to the Jacobian determinant in general relativity. Moreover, the linear relationship between Δxmin and the Planck length ℓp is established with a proportionality factor of −〈g〉μνβ2, where β2 denotes the RGUP parameter. The discretization of spacetime coordinates results in the discontinuity of the test particle’s wavefunction ψ(x,t), leading to an unrealistic Δp. It is also noted that the lower limit of 〈(pμ)2〉 is directly proportional to the fundamental tensor. [ABSTRACT FROM AUTHOR]

A new mechanism based on space-time noncommutativity has shown to produce anisotropy and an axial-like interaction, yielding leptonic asymmetry for fermionic particles propagating in a noncommutative curved FRW universe. As a byproduct, an analytical expression of this asymmetry for ultra-relativistic particles, such as neutrinos, is derived explicitly. Constraints and bounds from the cosmological parameters are also discussed. [ABSTRACT FROM AUTHOR]

We revisit the construction of quantum Riemannian geometries on graphs starting from a hermitian metric compatible connection, which always exists. We use this method to find quantum Levi-Civita connections on the n-leg star graph for n = 2 , 3 , 4 and find the same phenomenon as recently found for the A n Dynkin graph that the metric length for each inbound arrow has to exceed the length in the other direction by a multiple, here n . We then study quantum geodesics on graphs and construct these on the 4-leg graph and on the integer lattice line Z with a general edge-symmetric metric. [ABSTRACT FROM AUTHOR]

We solve for quantum geometrically realised pre-spectral triples or ‘Dirac operators’ on the noncommutative torus C θ [ T 2 ] and on the algebra M 2 (C) of 2 × 2 matrices with their standard quantum metrics and associated quantum Riemannian geometry. For C θ [ T 2 ] , we obtain a standard even spectral triple but now uniquely determined by full geometric realisability. For M 2 (C) , we are forced to a particular flat quantum Levi-Civita connection and again obtain a natural fully geometrically realised even spectral triple. In both cases there is an odd spectral triple for a different choice of a sign parameter. We also consider an alternate quantum metric on M 2 (C) with curved quantum Levi-Civita connection and find a natural 2-parameter family of Dirac operators which are almost spectral triples, where fails to be antihermitian. In all cases, we split the construction into a local tensorial level related to the quantum Riemannian geometry, where we classify the results more broadly, and the further requirements relating to the pre-Hilbert space structure. We also illustrate the Lichnerowicz formula for which applies in the case of a full geometric realisation. [ABSTRACT FROM AUTHOR]

Lie–Poisson gauge formalism provides a semiclassical description of noncommutative U(1) gauge theory with Lie algebra type noncommutativity. Using the Dirac approach to constrained Hamiltonian systems, we focus on a class of Lie–Poisson gauge models, which exhibit an admissible Lagrangian description. The underlying noncommutativity is supposed to be purely spatial. Analyzing the constraints, we demonstrate that these models have as many physical degrees of freedom as there are present in the Maxwell theory. [ABSTRACT FROM AUTHOR]

We study a Hermitian matrix model with a kinetic term given by T r (H Φ 2) , where H is a positive definite Hermitian matrix, similar as in the Kontsevich Matrix model, but with its potential Φ 3 replaced by Φ 4 . We show that its partition function solves an integrable Schrödinger-type equation for a non-interacting N-body Harmonic oscillator system. [ABSTRACT FROM AUTHOR]

Within the quantum field theory approach and using the technique of Bogoliubov transformations, the von Neumann boson-antiboson pair creation quantum entanglement entropy is studied in the context of the noncommutative Bianchi I universe. The investigations have shown that the behavior of entanglement entropy is strongly influenced by the choice of the noncommutativity θ parameter, k ⊥ -frequency modes and the structure of the curved spacetime. Moreover, the relationship between thermodynamics, Bose–Einstein condensation and quantum entanglement is also discussed. [ABSTRACT FROM AUTHOR]

Placing a Dirac--Schrödinger operator along the orbit of a flow on a compact manifoldM defines an R-equivariant spectral triple over the algebra of smooth functions on M. We study some of the properties of these triples, with special attention to their zeta functions. These zeta functions are defined for Re(s) > 1 by Trace(fpH-8), where fp is the uniformly continuous function on the real line obtained by restricting the continuous or smooth function f on M to the orbit of a point p ∈ M, and H = -∂²/∂x² + x² is the harmonic oscillator. The meromorphic continuation property and pole structure of these zeta functions are related to ergodic time averages in dynamics. In the case of the periodic flow on the circle, one obtains a spectral triple over the smooth irration torus A∞h ⊂ Ah already studied by Lesch and Moscovici. We strengthen a result of these authors, showing that the zeta function Trace(aH-8) extends meromorphically to C for any element a of the C*-algebra Ah. Another variant of our construction yields a spectral cycle for Ah ⊗ A1/h and a spectral triple over a suitable subalgebra with the meromorphic continuation property if h satisfies a Diophantine condition. The class of this cycle defines a fundamental class in the sense that it determines a KK-duality between Ah and A1/h. We employ the local index theorem of Connes and Moscovici in order to elaborate an index theorem of Connes for certain classes of differential operators on the line and compute the intersection form on K-theory induced by the fundamental class. [ABSTRACT FROM AUTHOR]

Despite the obvious progress made by the Feynman, Ravndal, and Kislinger relativistic model in describing the internal motion of a system with confinement of quarks in a nucleon, it turned out to be insufficiently realistic for a number of reasons. In particular, the model does not take into account some cornerstone properties of QCD, namely, gluon exchange between quarks, the influence of the resulting quark sea on valence quarks, and the self-interaction of colored gluons. It is these phenomena that spontaneously break the chiral symmetry of the quark system and form the bulk of the nucleon. To eliminate the above shortcomings of the model, the problem of self-organization of a three-quark dynamical system immersed in a colored quark–antiquark sea is considered within the framework of complex probabilistic processes that satisfy the stochastic differential equation of the Langevin–Kline–Gordon–Fock type. Taking into account the hidden symmetry of the internal motion of a dynamical system, a mathematically closed nonperturbative approach was developed, which makes it possible to construct the mathematical expectation of the wave function and other parameters of the nucleon in the form of multiple integral representations. It is shown that additional subspaces arising in a representation characterized by a noncommutative geometry with topological features participate in the formation of an effective interaction between valence quarks against the background of harmonic interaction between them. [ABSTRACT FROM AUTHOR]

We study a real symmetric Φ 4 -matrix model whose kinetic term is given by Tr (E Φ 2) , where E is a positive diagonal matrix without degenerate eigenvalues. We show that the partition function of this matrix model corresponds to a zero-energy solution of a Schrödinger type equation with Calogero–Moser Hamiltonian. A family of differential equations satisfied by the partition function is also obtained from the Virasoro algebra. [ABSTRACT FROM AUTHOR]

In this paper, we discuss exact realizations of the Yang–Poisson model on canonical phase space. The Yang model is an example of noncommutative geometry on a background space–time of constant curvature and is notable for its duality between position and momentum manifolds. We call Yang–Poisson model its classical limit, with commutators replaced by Poisson brackets. The structure is simpler in the classical case, and exact realizations can be found. [ABSTRACT FROM AUTHOR]

We formulate a notion of jet bundles over a possibly noncommutative algebra A equipped with a torsion-free connection. Among the conditions needed for third-order jets and above is that the connection also be flat and its ‘generalised braiding tensor’ σ : Ω 1 ⊗ A Ω 1 → Ω 1 ⊗ A Ω 1 obey the Yang–Baxter equation or braid relations. We also cover the case of jet bundles of a given ‘vector bundle’ over A in the form of a bimodule E with a flat bimodule connection with its braiding σ E obeying the coloured braid relations. Examples include the permutation group S 3 with its 2-cycles calculus, M 2 (C) , the bicrossproduct model quantum spacetime in two dimensions and C q [ S L 2 ] with its 3-dimentional left covariant calculus and q a 4th root of unity. [ABSTRACT FROM AUTHOR]

We study a noncommutative (NC) deformation of a charged scalar field, minimally coupled to a classical (commutative) Reissner–Nordström-like background. The deformation is performed via a particularly chosen Killing twist to ensure that the geometry remains undeformed (commutative). An action describing a NC scalar field minimally coupled to the RN geometry is manifestly invariant under the deformed U (1) ⋆ gauge symmetry. We find the equation of motion and conclude that the same equation is obtained from the commutative theory in a modified geometrical background described by an effective metric. This correspondence we call "duality between formal and effective approach". We also show that a NC deformation via semi-Killing twist operator cannot be rewritten in terms of an effective metric. There is dual description for those particular deformations. [ABSTRACT FROM AUTHOR]

In this contribution, we summarize our recent studies of the phase structure of the Grosse–Wulkenhaar model and its connection to renormalizability. Its action contains a special term that couples the field to the curvature of the noncommutative background space. We first analyze the numerically obtained phase diagram of the model and its three phases: the ordered, the disordered, and the noncommutative stripe phase. Afterward, we discuss the analytical derivation of the effective action and the ordered-to-stripe transition line, and how the obtained expression successfully explains the curvature-induced shift of the triple point compared to the model without curvature. This shift also causes the removal of the stripe phase and makes the model renormalizable. [ABSTRACT FROM AUTHOR]

It has been claimed that wormholes are just as good a prediction of Einstein’s theory as black holes, but they are subject to severe restrictions from quantum field theory. The purpose of this paper is to show that the claim can be substantiated in spite of these restrictions.

Relativistic and nonrelativistic quantum mechanics formulated in a noncommutative space–space have recently become the object of renewed interest. In the context of three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries with arbitrary spin-orbit coupling quantum number k , we approximate to solve the deformed Dirac equation for a new suggested new generalized Morse potential and a class of Yukawa potential including improved Coulomb-like tensor interaction (N(GMP-CYP) plus ICLTI). In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the global new energy eigenvalue which equals the energy eigenvalue in the usual relativistic QM as the main part plus three corrected parts produced from the effect of the spin-orbit interaction, the new modified Zeeman, and the rotational Fermi term, by using the parametric of the well-known Bopp's shift method and standard perturbation theory using Greene–Aldrich approximation to nonlinear and exponential terms in the effective potential. The new values that we got appeared sensitive to the quantum numbers (j , k , l / l ̃ , s / s ̃ , m / m ̃), the mixed potential depths ( D e , r e , a , b , c), the range of the potential α and noncommutativity parameters (Θ , τ , χ). We have studied the nonrelativistic limit of new spin symmetry under the N(GMP-ICYP) model, we will also treat some important special cases such as the new generalized Morse potential, the new class of Yukawa potential, the new Hellmann potential, the new inversely quadratic Yukawa potential, the new Hulthén potential and new Coulomb potential. Finally, we studied a case of composite systems. [ABSTRACT FROM AUTHOR]

We investigate a quantum mechanical harmonic oscillator based on the extended Snyder model. This realization of the Snyder model is constructed as a quantum phase space generated by D spatial coordinates and D (D − 1) / 2 tensorial degrees of freedom, together with their conjugated momenta. The coordinates obey nontrivial commutation relations and generate a noncommutative geometry, which admits nicer properties than the usual realization of the model, in particular giving rise to an associative star product. The spectrum of the harmonic oscillator is studied through the introduction of creation and annihilation operators. Some physical consequences of the introduction of the additional degrees of freedom are discussed. [ABSTRACT FROM AUTHOR]

The main findings of the generalized uncertainty principle (GUP), the phenomenological approach, for instance, the emergence of a minimal measurable length uncertainty, are obtained in various versions from theories of quantum gravity, such as string theory, loop quantum gravity, doubly special relativity and black hole physics. GUP counts for impacts of relativistic energies and finite gravitational fields on the fundamental theories of quantum mechanics (QM), the noncommutation and measurement uncertainty. Utilizing GUP in reconciling principles of general relativity (GR) and QM, thereby enables to draw convincing conclusions about quantum gravity. To resolve the shortcuts reported with the nonrelativistic three-dimensional GUP, namely, violation of Lorentz covariance, dependence on frame of reference, and violation of the linear additional law of momenta, we introduce relativistic four-dimensional generalized uncertainty principle (RGUP) to curved spacetime. To unify GR and QM, we apply the Born reciprocity principle (BRP), distance-momentum duality symmetry and RGUP to estimate the fundamental tensor in discretized curved spacetime. To this end, we generalize Riemann geometry. The Finsler geometry, which is characterized by manifold and Finsler structure, allows to directly apply RGUP to the Finsler structure of a free particle so that F ( x ̂ 0 , ẋ 0) can be expressed as F ( x ̂ 0 , ϕ ̂ p ̂ 0) , from which the metric tensor in discretized Riemann spacetime could be deduced. We conclude that F ( x ̂ 0 , ϕ ̂ p ̂ 0) is homogeneous with degree 1 in p ̂ 0 , while ϕ ̂ = 1 + β 1 p ̂ 0 ρ p ̂ 0 ρ is 0 -homogeneous resulting in g ̃ μ ν ≃ (+ 2 β 1 F 2) [ 1 + m ̄ 2 | ẍ ̂ | 2 ] g μ ν . Despite, the astonishing similarity with the conformal transformation, know as Weyl tensor, this study suggests that principles of QMs could be unambiguously imposed on the resulting fundamental tensor. Also, we conclude that the features of Finsler geometry assumed in this study are likely the ones of the duel Hamilton geometry. [ABSTRACT FROM AUTHOR]

We discuss various aspects of noncommutative geometry of smooth subalgebras of Bunce-Deddens-Toeplitz algebras. [ABSTRACT FROM AUTHOR]

In this paper we study ensembles of finite real spectral triples equipped with a path integral over the space of possible Dirac operators. In the noncommutative geometric setting of spectral triples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus, this path integral serves as a noncommutative analogue of integration over metrics, a key feature of a theory of quantum gravity. From these integrals in the so-called double scaling limit we derive critical exponents of minimal models from Liouville conformal field theory coupled with gravity. Additionally, the asymptotics of the partition function of these models satisfy differential equations such as Painlevé I, as a reduction of the KDV hierarchy, which is predicted by conformal field theory. This is all proven using well-established and rigorous techniques from random matrix theory. [ABSTRACT FROM AUTHOR]

We study geodesics flows on curved quantum Riemannian geometries using a recent formulation in terms of bimodule connections and completely positive maps. We complete this formalism with a canonical ∗ operation on noncommutative vector fields. We show on a classical manifold how the Ricci tensor arises naturally in our approach as a term in the convective derivative of the divergence of the geodesic velocity field and use this to propose a similar object in the noncommutative case. Examples include quantum geodesic flows on the algebra of 2 × 2 matrices, fuzzy spheres and the q-sphere. [ABSTRACT FROM AUTHOR]

We review the present status of gauge theories built on various quantum space–times described by noncommutative space–times. The mathematical tools and notions underlying their construction are given. Different formulations of gauge theory models on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra are covered, with particular emphasis on some explored quantum properties. Recent attempts aiming to include gravity dynamics within a noncommutative framework are also considered. [ABSTRACT FROM AUTHOR]

We study the duals of a certain class of finite-dimensional operator systems, namely the class of operator systems associated to tolerance relations on finite sets or equivalently the class of operator systems that are associated with graphs. In the case where the graphs associated with these operator systems are chordal we are able to find concrete realizations of their duals as sitting inside of finite-dimensional C ∗ -algebras. We then use these concrete realizations to compute the C ∗ -envelopes, propagation numbers and extremal rays of these duals in the chordal case. Finally, we exemplify our results by applying them to operator systems of band matrices. [ABSTRACT FROM AUTHOR]

The structure of the proper time of both massive and massless particles is considered in the framework of the pseudoclassical model of supertime. Variants of the introduction of non-commutative geometry into the structure of supertime for the models of fundamental particles described in previous publications are proposed. [ABSTRACT FROM AUTHOR]

We construct the novel Clifford-Yang algebra which is an extension of the Yang algebra in noncommutative phase spaces. The Clifford-Yang algebra allows us to write down the commutators of the n o n c o m m u t a t i v e polyvector-valued coordinates and momenta which are compatible with the Jacobi identities, the Weyl–Heisenberg algebra, and paves the way for a formulation of quantum mechanics (QM) in noncommutative Clifford spaces. We continue with a detailed study of the isotropic 3D quantum oscillator in noncommutative spaces and find the energy eigenvalues and eigenfunctions. These findings differ considerably from the ordinary quantum oscillator in commutative spaces. We find that QM in noncommutative spaces leads to very different solutions, eigenvalues, and uncertainty relations than ordinary QM in commutative spaces. The generalization of QM to noncommutative Clifford (phase) spaces is attained via the Clifford-Yang algebra. The operators are now given by the generalized angular momentum operators involving polyvector coordinates and momenta. The eigenfunctions (wave functions) are now more complicated functions of the polyvector coordinates. We conclude with some important remarks. [ABSTRACT FROM AUTHOR]

This paper deals with some questions regarding the notion of integral in the framework of Connes' noncommutative geometry. First, we present a purely spectral theoretic construction of Connes' integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue's integral. This answers another question of Alain Connes. We further clarify the relationship of Connes' integration with Weyl's laws for compact operators and Birman-Solomyak's perturbation theory. We also give a "soft proof" of Birman-Solomyak's Weyl's law for negative order pseudodifferential operators on closed manifold. This Weyl's law yields a stronger form of Connes' trace theorem. Finally, we explain the relationship between Connes' integral and semiclassicalWeyl's law for Schrödinger operators, including for (fractional) Schrödinger operators on Euclidean spaces and on noncommutative manifolds. We thus get a neat links between noncommutative geometry and semiclassical analysis. [ABSTRACT FROM AUTHOR]

When gravity is quantum, the point structure of space-time should be replaced by a non-commutative geometry. This is true even for quantum gravity in the infra-red. Using the octonions as space-time coordinates, we construct pre-spacetime, pre-quantum Lagrangian dynamics. We show that the symmetries of this non-commutative space unify the standard model of particle physics with SU(2)R chiral gravity. The algebra of the octonionic space yields spinor states which can be identified with three generations of quarks and leptons. The geometry of the space implies quantisation of electric charge, and leads to a theoretical derivation of the mysterious mass ratios of quarks and the charged leptons. Quantum gravity is quantisation not only of the gravitational field, but also of the point structure of space-time.

We use the noncommutative Higgs effective standard model to make a phenomenological prediction for the transverse momentum distribution of the Higgs boson produced in association with a jet at hadron colliders. We calculate at leading order in the noncommutative parameter Θ as well as leading order in the strong coupling α s , the one-loop p t distribution of the Higgs boson. As in the standard model, the fixed-order distribution suffers from large logarithms at small p t which require an all-orders resummation. We find that the large- p t region of the distribution is strongly affected by the non-commutativity, while small- p t region is not. Following this observation, we propose a simple matching method that allows us to compute a result that is also valid at small p t obtained with standard-model parton showers such as Pythia 8. We also compare our results with the next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) distributions in the standard model, in order to assess the importance of higher-order effects in the search for non-commutativity at colliders. [ABSTRACT FROM AUTHOR]

Relativistic and nonrelativistic quantum mechanics formulated in a noncommutative space-space have recently become the object of renewed interest. In the context of extended relativistic quantum mechanics (ERQM) symmetries with arbitrary spin-orbit coupling quantum number k , we approximate to solve the deformed Dirac equation (DDE) for a new suggested new generalized Schiöberg and Manning–Rosen potentials within the generalized (Coulomb and Yukawa)-like tensor interactions (NGSM-GLTs). In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the global new energy eigenvalue which equals the energy eigenvalue in usual relativistic quantum mechanics (RQM) as the main part plus three corrected parts produced from the effect of the spin-orbit interaction, the new modified Zeeman, and the rotational Fermi term, by using the parametric of the well-known Bopp's shift method and standard perturbation theory using Greene–Aldrich approximation to handle 1 / r , 1 / r 2 and other terms in the effective potential. The new values that we got appeared sensitive to the quantum numbers (j , k , l / l ̃ , s / s ̃ , m / m ̃), the mixed potential depths ( D 0 , D 1 , σ , α , A), the range of the potential δ , and noncommutativity parameters (Θ , τ , χ). We recovered several potentials, including the improved Schiöberg and Manning–Rosen potentials within the improved Yukawa-like tensor interaction, the new Schiöberg and Manning–Rosen potentials within the improved Coulomb-like tensor interaction, the new Schiöberg potential within the improved Yukawa-like tensor interaction, the new Manning–Rosen potential within the improved Yukawa-like tensor interaction, and the new Schiöberg and Manning–Rosen potentials potential problems in the context of nonrelativistic extended quantum mechanics symmetries. [ABSTRACT FROM AUTHOR]

In the context of extended relativistic quantum mechanics (ERQM) symmetries with arbitrary spin–orbit coupling quantum numbers k, we approximate solve the deformed Dirac equation for a new suggested improved Schiöberg potential within Yukawa-like tensor interaction (ISPYLT). In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the global new energy eigenvalue which equals the energy eigenvalue, in the usual relativistic QM, plus the corrected energy induced by three infinitesimal additive parts of the Hamiltonian, corresponding to the spin–orbit interaction, the new modified Zeeman, and the rotational Fermi term, by using the parametric of Bopp's shift method and standard perturbation theory. The new values that we got appeared sensitive to the quantum numbers j , k , l , l p , s , s p , m , m p , the mixed potential depths ( D 0 , σ , V 0 ), the range of the potential α , and noncommutativity parameters. We have also treated the nonrelativistic limit of the studied potential in NR-EQM symmetries. It is worthwhile to mention that for the simultaneous limits η , β , χ → 0 , 0 , 0 , we recover the equations of energy for the spin symmetry and the p-spin symmetry, under the Schiöberg potential, including a Yukawa-like tensor interaction which is known in the literature. [ABSTRACT FROM AUTHOR]

The Arens-Michael functor in noncommutative geometry is an analogue of the analytification functor in algebraic geometry: out of the ring of "algebraic functions" on a noncommutative affine scheme, it constructs the ring of "holomorphic functions" on it when viewed as a noncommutative complex analytic space. In this paper, we explicitly compute the Arens-Michael envelopes of the Jordan plane and the quantum enveloping algebra Uq(sl)2)) of sl(2) for |q| = 1. [ABSTRACT FROM AUTHOR]

We take a fresh look at the geometrization of logic using the recently developed tools of "quantum Riemannian geometry" applied in the digital case over the field F2 = {0, 1}, extending de Morgan duality to this context of differential forms and connections. The 1-forms correspond to graphs and the exterior derivative of a subset amounts to the arrows that cross between the set and its complement. The line graph 0 - 1 - 2 has a non-flat but Ricci flat quantum Riemannian geometry. The previously known four quantum geometries on the triangle graph, of which one is curved, are revisited in terms of left-invariant differentials, as are the quantum geometries on the dual Hopf algebra, the group algebra of Z3. For the square, we find a moduli of four quantum Riemannian geometries, all flat, while for an n-gon with n > 4 we find a unique one, again flat. We also propose an extension of de Morgan duality to general algebras and differentials over F2. [ABSTRACT FROM AUTHOR]

Noncommutative geometry, an offshoot of string theory, replaces point-like particles with smeared objects. These local effects have led to wormhole solutions in a semiclassical setting, but it has also been claimed that the noncommutative effects can be implemented by modifying only the energy-momentum tensor in the Einstein field equations, while leaving the Einstein tensor unchanged. The implication is that noncommutative-geometry wormholes could be macroscopic. The purpose of this paper is to confirm this conclusion in a simpler and more concrete manner by showing that the throat radius can indeed be macroscopic. This result can be readily explained by considering the noncommutative-geometry background to be a fundamental property and the macroscopic wormhole spacetime to be emergent.

Noncommutative Kähler structures were recently introduced as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant Kähler structures of irreducible type (those having an irreducible space of holomorphic 1-forms) we provide simple cohomological criteria for positivity, allowing one to avoid explicit curvature calculations. These general results are applied to our motivating family of examples, the irreducible quantum flag manifolds O q (G / L S) . Building on the recently established noncommutative Borel–Weil theorem, every relative line module over O q (G / L S) can be identified as positive, negative, or flat, and it is then concluded that each Kähler structure is of Fano type. [ABSTRACT FROM AUTHOR]

We build on our construction of Hopf algebroids from noncommutative calculi under the further assumption of surjectivity for the calculus. We also introduce the notions of Hopf ideals and isotropy quotients for arbitrary Hopf algebroids. Using these ingredients, we prove a Riemann-Hilbert correspondence for digraphs, by showing that the groupoid algebra of the fundamental groupoid of a digraph is isomorphic to the isotropy quotient of the Hopf algebroid corresponding to flat connections over the digraph. [ABSTRACT FROM AUTHOR]

We review the construction of the λϕ 4-model on noncommutative geometries via exact solutions of Dysonâ€"Schwinger equations and explain how this construction relates via (blobbed) topological recursion to problems in algebraic and enumerative geometry. [ABSTRACT FROM AUTHOR]

We review recent progress in the analytic study of random matrix models suggested by noncommutative geometry. One considers fuzzy spectral triples where the space of possible Dirac operators is assigned a probability distribution. These ensembles of Dirac operators are constructed as toy models of Euclidean quantum gravity on finite noncommutative spaces and display many interesting properties. The ensembles exhibit spectral phase transitions, and near these phase transitions they show manifold-like behavior. In certain cases one can recover Liouville quantum gravity in the double scaling limit. We highlight examples where bootstrap techniques, Coulomb gas methods, and Topological Recursion are applicable. [ABSTRACT FROM AUTHOR]

Based on the perturbation expansion, we compute the noncommutative corrections to the minimal surface areas of the pure AdS spacetime and Schwarzschild-AdS black hole, where the noncommutative background is suitably constructed in terms of the Poincaré coordinate system. In particular, we find a reasonable tetrad with subtlety, which not only matches the metrics of the pure AdS spacetime and Schwarzschild-AdS black hole in the commutative case, but also makes the corrections real rather than complex in the noncommutative case. For the pure AdS spacetime, the nocommutative effect is only a logarithmic term, while for the Schwarzschild-AdS black hole, it contains a logarithmic contribution plus both a mass term and a noncommutative parameter related term. Furthermore, we show that the holographic entanglement entropy with noncommutativity obeys a relation which is similar to the first law of thermodynamics in the pure AdS spacetime. [ABSTRACT FROM AUTHOR]

In the setting of non-type II 1 representations, we propose a definition of deformed Fredholm module [ D T | D T | - 1 , · ] T for a modular spectral triple T , where D T is the deformed Dirac operator. D T is assumed to be invertible for the sake of simplicity, and its domain is an “essential” operator system E T . According to such a definition, we obtain [ D T | D T | - 1 , · ] T = | D T | - 1 d T (·) + d T (·) | D T | - 1 , where d T is the deformed derivation associated to D T . Since the “quantum differential” 1 / | D T | appears in a symmetric position, such a definition of Fredholm module differs from the usual one even in the undeformed case, that is in the tracial case. Therefore, it seems to be more suitable for the investigation of noncommutative manifolds in which the nontrivial modular structure might play a crucial role. We show that all models in Fidaleo and Suriano (J Funct Anal 275:1484–1531, 2018) of non-type II 1 representations of noncommutative 2-tori indeed provide modular spectral triples, and in addition deformed Fredholm modules according to the definition proposed in the present paper. Since the detailed knowledge of the spectrum of the Dirac operator plays a fundamental role in spectral geometry, we provide a characterisation of eigenvalues and eigenvectors of the deformed Dirac operator D T in terms of the periodic solutions of a particular class of eigenvalue Hill equations. [ABSTRACT FROM AUTHOR]

A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion incorporates familiar examples like fuzzy spheres and fuzzy tori. In the framework of random noncommutative geometry, we use Barrett's characterization of Dirac operators of fuzzy geometries in order to systematically compute the spectral action S.D/ D Tr f .D/ for 2n-dimensional fuzzy geometries. In contrast to the original Chamseddine-- Connes spectral action, we take a polynomial f with f .x/ ! 1 as jxj ! 1 in order to obtain a well-defined path integral that can be stated as a random matrix model with action of the type S.D/ D N tr F CPi tr Ai tr Bi, being F, Ai and Bi noncommutative polynomials in 22n are obtained via chord diagrams and satisfy: independence of N; self-adjointness of the main polynomial F (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of Ai and Bi simultaneously, for fixed i . Collectively, this favors a free probabilistic perspective for the large-N limit we elaborate on. [ABSTRACT FROM AUTHOR]

We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) L p -spaces associated to finite measure spaces. More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension d exceeds the spectral dimension, i.e. that the associated Hodge-Dirac operator is d + -summable. We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups. Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge. Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective. [ABSTRACT FROM AUTHOR]

We study the Laplacian operator Δ ∂ ‾ associated to a Kähler structure (Ω (• , •) , κ) for the Heckenberger–Kolb differential calculus of the quantum quadrics O q (Q N) , which is to say, the irreducible quantum flag manifolds of types B n and D n. We show that the eigenvalues of Δ ∂ ‾ on zero forms tend to infinity and have finite multiplicity. [ABSTRACT FROM AUTHOR]

We introduce ringed spaces, referred to as para-manifolds, whose non-commutative nilpotent coordinates naturally describe parafermions at the classical level in a similar way as Grassmann variables describe usual fermions. Given a supermanifold X , we construct a family of para-manifolds X (p) for positive integers p , such that X (1) recovers the supermanifold itself. A differential analysis on para-manifolds is developed, which can be readily applied to model physical problems. Two classes of para-manifolds, respectively corresponding to X being affine superspaces and projective superspaces, are treated in detail as examples. Green's theory of parafermions is reformulated in terms of para-manifolds. [ABSTRACT FROM AUTHOR]

We investigate a quantum mechanical harmonic oscillator based on the extended Snyder model. This realization of the Snyder model is constructed as a quantum phase space generated by D spatial coordinates and D(D−1)/2 tensorial degrees of freedom, together with their conjugated momenta. The coordinates obey nontrivial commutation relations and generate a noncommutative geometry, which admits nicer properties than the usual realization of the model, in particular giving rise to an associative star product. The spectrum of the harmonic oscillator is studied through the introduction of creation and annihilation operators. Some physical consequences of the introduction of the additional degrees of freedom are discussed.

The Wilsonian renormalization group approach to matrix models is outlined and applied to multitrace matrix models with an emphasis on the computation of the fixed points which could describe the phase structure of noncommutative scalar phi-four theory. [ABSTRACT FROM AUTHOR]

We give the superalgebra of N = 2 chiral (and antichiral) quantum superfields realized as a subalgebra of the quantum supergroup SL q (4|2). The multiplication law in the quantum supergroup induces a coaction on the set of chiral superfields. We also realize the quantum deformation of the chiral Minkowski superspace as a quantum principal bundle. [ABSTRACT FROM AUTHOR]

In this paper, the wormholes have been constructed in the framework of the linear f (R , T) gravity model with noncommutative geometry. The wormhole solutions are obtained in the context of Gaussian and Lorentzian distributions of matter using f (R , T) gravity. The energy conditions such as null, weak, strong, and dominant energy conditions are explored and the regions satisfying these conditions are examined for both forms of matter distributions. Further, the stability of wormhole solutions is studied with the help of TOV (Tolman–Oppenheimer–Volkoff) equations. It is found that the wormhole solutions are stable and free from exotic matter for the suitable choice of parameters. [ABSTRACT FROM AUTHOR]

When gravity is quantum, the point structure of space-time should be replaced by a non-commutative geometry. This is true even for quantum gravity in the infra-red. Using the octonions as space-time coordinates, we construct pre-spacetime, pre-quantum Lagrangian dynamics. We show that the symmetries of this non-commutative space unify the standard model of particle physics with S U (2) R chiral gravity. The algebra of the octonionic space yields spinor states which can be identified with three generations of quarks and leptons. The geometry of the space implies quantisation of electric charge, and leads to a theoretical derivation of the mysterious mass ratios of quarks and the charged leptons. Quantum gravity is quantisation not only of the gravitational field, but also of the point structure of space-time. [ABSTRACT FROM AUTHOR]